Since all of our speculation is so limited by us being unsure as to what sort of regime this White Russia is, let's set up a scenario and go from there.
I'll start with my favorite "White Victory" scenario:
Denikin wins the Battle of Oryol in October of 1919 and advances on Moscow. The Bolshies pull back the reinforcements they sent to relieve Petrograd and that city falls to Yudenich later that month. Moscow then falls to the Whites a bit later.
At this point, the top White leaders are Denikin, Yudenich, Wrangel, and Kolchak, all of who have (to varying degrees) both monarchist and reformist sympathies. They run the country as a junta until mid-late 1920 when they elect Grand Duke Nicholas as Tsar since the country needs a popular figurehead to unite around. Nicholas III is a relatively powerless constitutional monarch, something made clear from day one.
A Duma is set up, Wrangel or Milyukov is chosen as PM, and chances are we have sort of Weimar-like situation evolve with a chaotic political scene and the army (probably headed by Denikin) ready to step in if things get too out of hand.
Aaaaand....
GO!